**Core Concept**
The question tests the understanding of hepatitis B virus (HBV) transmission from mother to child, particularly in the context of acute liver failure in an infant. The underlying principle is the serological profile of a hepatitis B surface antigen (HbsAg) carrier mother and its implications for vertical transmission.

**Why the Correct Answer is Right**
In vertical transmission, HBV is transmitted from mother to child during birth. An HbsAg-positive mother can have one of two serological profiles: (i) the mother is an HbsAg carrier with a negative hepatitis B e-antigen (HBeAg) and high levels of hepatitis B surface antibody (HbsAb), indicating a low viral load and a lower risk of transmitting the virus to the child; or (ii) the mother is an HbsAg carrier with a positive HBeAg and low levels of HbsAb, indicating a high viral load and a higher risk of transmitting the virus to the child. In this case, the infant developed acute liver failure two days after birth, suggesting a high viral load and a high risk of transmission. Therefore, the mother's serological profile is likely to be HbsAg-positive with a positive HBeAg and low levels of HbsAb.

**Why Each Wrong Option is Incorrect**
**Option A:** This option is incorrect because a mother who is only HbsAb-positive is not a carrier and does not pose a risk of transmitting the virus to the child.

**Option B:** This option is incorrect because a mother who is HbsAg-negative and HbsAb-positive is not a carrier and does not pose a risk of transmitting the virus to the child.

**Option D:** This option is incorrect because a mother who is HbsAg-negative and HBeAg-negative is not a carrier and does not pose a risk of transmitting the virus to the child.

**Clinical Pearl / High-Yield Fact**
In infants born to HbsAg-positive mothers, the risk of transmission is higher if the mother is also positive for HBeAg. This is because HBeAg indicates a high viral load and increased infectivity.

**Correct Answer: C**